Size-dependent bending behavior of three-layered doubly curved shells: Modified couple stress formulation

Abstract

In this paper, modified couple stress formulation of a small scale doubly curved piezoelectric shell resting on Pasternak's foundation is presented based on first-order shear deformation theory. Size-dependent electro-elastic results of doubly curved shell are presented based on an analytical approach. The doubly curved piezoelectric shell is subjected to uniform transverse loads and applied voltage. To account the size dependency, modified couple stress theory is employed in conjunction with principle of virtual work. The numerical results are presented in both tabular and graphical forms to show the influence of small scale parameter, applied voltage, geometries and two parameters of Pasternak's foundation on the electro-elastic results of size-dependent doubly curved piezoelectric shell.

Keywords

Size-dependent doubly curved piezoelectric shells small scale parameter applied voltage first-order shear deformation theory Winkler-Pasternak's Foundation

Introduction

Analysis of structures in very small scale (micro or nano scales) has enforced researchers to find new non-classical theories to cover prediction of behavior of those in various environments. These theories were presented to account size-dependencies in the constitutive relations. Advances in development of non-classical theories lead to various theories in micro and nano scales for better prediction of behavior of small scale structures. Nonlocal Eringen elasticity theory, modified couple stress theory, strain gradient theory, and nonlocal strain gradient theory were proposed for the analysis of structures in nano and micro scales. Although the application of above mentioned theories on the custom structures such as rods, beams, and plates has been presented by various researchers, analysis of non-flat structures such as curved beam and doubly curved shell has not been performed comprehensively. Literature review on the subject of this paper is presented to justify the novelties and necessities of this study.

The influence of various aspect ratios and imperfections was studied on the post-buckling analysis of an imperfect doubly curved shell by Kapania and Yang [1]. Fan and Zhang [2] used curvilinear coordinate system to study the static and dynamic analysis of the simply-supported orthotropic doubly curved shells based on a unified analytical solution for thin, moderately thick, and thick laminated shells. Tadi Beni et al. [3] used modified couple stress theory for free vibration analysis of functionally graded cylindrical thin shell with various boundary conditions based on Hamilton's principle. The influence of important parameters such as length scale parameter, in-homogeneous index and some dimensionless geometric parameters have been studied on the responses of cylindrical shell. Arefi et al. [4] studied the size-dependent free vibration analysis of a three-layered exponentially graded nano-/micro-plate with piezomagnetic face sheets resting on Pasternak’s foundation based on Hamilton's principle and first-order shear deformation theory. The modified couple stress theory was used to account size dependency in the formulations. Free vibration analysis of a functionally graded cylindrical shell was studied by Tadi Beni et al. [5] based on shear deformation theory and modified couple stress theory. The material properties have been graded along the thickness direction based on power law distribution using an in-homogeneous index. Size-dependent analysis was performed using micro length scale parameter associated with modified couple stress theory. Size-dependent free vibration analysis of three-layered exponentially graded nanoplate with piezomagnetic face-sheets resting on Pasternak’s foundation was studied based on nonlocal elasticity theory and first-order shear deformation theory [6]. Free vibration analysis of laminated composite doubly curved shells was studied by Chandrashekhara [7] based on first-order shear deformation theory. Influence of in-plane and rotary inertia was accounted in the elements of mass matrix. Arefi and Zenkour [8] studied the influence of thermo-magneto-electro-mechanical loads on the static analysis of a three-layered nanoplate. Sinusoidal shear-deformation plate theory was used for the formulation of problem and principle of virtual displacement was employed for derivation of the governing equations. A complete set of equations for electro-magneto-elastic analysis of a thick shell of revolution with arbitrary thickness and curvature were derived based on magneto-electro-elastic relations [9].

Qatu and Asadi [10] presented the free vibration analysis of a thin shallow shell with various boundary conditions based on Ritz method. The influence of various parameters such as different boundary conditions and radii of curvature was studied on the responses. Size-dependent free vibration analysis of orthotropic doubly curved shallow shells with simply-supported was studied by Ghavanloo and Fazelzadeh [11] based on strain gradient theory and Novozhilov's linear shallow shell theory. The various length scale parameters were employed based on strain gradient theory for better prediction of behavior of small scale structure. Shooshtari and Razavi [12] studied the nonlinear and linear free vibration analyses of laminated magneto-electro-elastic doubly curved thin shell with simply-supported curved edges resting on an elastic foundation based on Donnell shell theory. The numerical results were calculated based on Lindstedt-Poincare perturbation method. The influence of parameters of foundation, geometrical characteristics, and electric and magnetic potentials was studied on the linear and nonlinear behaviors of these smart shells. Nonlinear vibrations of doubly curved cross-ply shells with simply-supported boundary conditions were studied by Yazdi [13] based on von-Karman geometric nonlinear theory with Donnell's shell equations. The nonlinear governing equations of motion were reduced to a second-order nonlinear ordinary differential equation using Galerkin approach and then solved using homotopy perturbation method. Wave propagation analysis of functionally graded piezoelectric nanorod excited to applied voltage was studied based on surface elasticity and nonlocal elasticity [14]. The effect of in-homogeneous index, applied voltage, and residual stresses was studied on the responses.

Zeighampour and Tadi Beni [15] employed first-order shear deformation theory for free vibration analysis of a cylindrical shell using modified couple stress theory. The numerical results have been verified through comparison with existing literature. They found that employing the small scale parameter based on modified couple stress theory leads to greater natural frequencies due to greater stiffness. Static, dynamic and free vibration analysis of FG doubly curved shells was studied by Tornabene et al. [16,17] based on various shear deformation theory and generalized differential quadrature method (GDQM). Tornabene et al. [18] studied the influence of carbon nanotube agglomeration on the free vibrations of laminated composite doubly curved shells and panels. Arefi and Zenkour [19] studied the free vibration, wave propagation, and tension analysis of functionally graded micro and nanorods based on nonlocal elasticity theory. The effect of applied electric potential was studied on the responses in detail. Tornabene et al. [20] studied free vibration analysis of doubly curved shells made of composite materials based on moving least squares differential quadrature (MLSDQ) method. The effect of volume fraction of carbon nanotubes, thickness ratio, aspect ratio, curvature ratio, and shallowness ratio was studied on the vibrational analysis of doubly curved FG composite panels reinforced with carbon nanotube by Pouresmaeeli and Fazelzadeh [21] based on ﬁrst-order shear deformation theory and Galerkin’s method. Arefi and Zenkour [22] studied the wave propagation analysis of a functionally graded piezoelectric Love nanorod model, based on employing the coupled stress components and surface elasticity. Zhou and Wang [23] presented the free vibration analysis of a micro cylindrical shell with simply-supported boundary condition based on modified couple stress theory. The influence of velocity of fluid flow and small scale parameter was studied on the responses. They showed that fluid inside the shell leads to decrease of natural frequencies with respect to case with no fluid.

Arefi and Zenkour [24] used sinusoidal shear deformation theory for thermo-magneto-electro-elastic analysis of a three-layered curved nanobeam. Nonlocal elasticity relations and Hamilton's principle were employed for derivation of the governing equations of motion. They mentioned that applied electric and magnetic potential leads to important changes of responses. Bending analysis of sandwich curved nanobeam was presented by Arefi and Zenkour [25]. The sandwich structure was made from a nano core and two piezomagnetic face-sheets. Influence of nonlocal parameter, applied electric and magnetic potentials, and two parameters of Pasternak's foundation was studied on the responses of the system. The numerical results indicate that increase of nonlocal parameters leads to decrease of stiffness of structure. Static analysis of a single layered curved nano beam was studied based on higher order shear deformation theory by Arefi and Zenkour [26]. Deformation and stress analysis of the curved nano beam were performed in terms of thermal loads and two parameters of foundation. Magneto-electro-elastic analysis of sandwich curved beam was studied by Arefi and Zenkour [27]. Zeighampour and Shojaeian [28] presented the buckling analysis of functionally graded sandwich cylindrical microshell subjected to axial loads based on couple stress theory and Donnel shell theory. Hamilton's principle was used to derive governing equations of motion for various boundary conditions. The numerical results were presented to examine the influence of material length scale parameter on the stability characteristics of the problem. The influence of applied electric potential and small scale parameter was studied on the free vibration characteristics of sandwich plates and beams [29,30].

Third-order shear and normal deformation theory were employed by Shah and Batra [31] for stretching and bending analyses of doubly curved shell. The numerical results were presented to consider the influence of various geometric parameters such as curvilinear length/thickness ratio, a/h, radius of curvature/curvilinear length, R/a, and the ratio of the two principal radii on the results. Kar et al. [32] studied the buckling analysis of a functionally graded doubly curved shell subjected to thermal loads based on finite element method. Gradation of material properties was accounted based on Voigt model. Liew and Lim [33] employed third-order shear deformation theory for vibration analysis of thick doubly curved shallow shell based on orthogonal curvilinear coordinate system. Coşkun [34] studied the influence of two-dimensional tensionless Pasternak foundation on the elastic response of an elastic beam subjected to harmonic load. Szekrényes [35] studied the analytical delamination of compliance and the energy release rate for the actual configuration based on an improved analysis. The effect of magneto-electric fields was studied on the wave propagation responses of functionally graded piezo-magnetic nanorod. Wave propagation analysis of a functionally graded magneto-electro-elastic nano-rod was studied by using nonlocal elasticity model subjected to electric and magnetic potentials. The influence of size-dependency was included in governing equations of motion [36].

A comprehensive literature review on the subject of this paper including various analyses of doubly curved shells and size-dependent theories was performed. This review indicates that there is no published work on the modified couple stress formulation of size-dependent doubly curved piezoelectric shell subjected to uniform transverse mechanical loads and applied electric potential. First-order shear deformation theory and couple stress formulations are used to derive electro-elastic governing relations based on six unknown variables including five displacement and rotation components and one electric potential. The numerical results are derived based on an analytical approach. The influence of small scale parameter, applied voltage, and two parameters of foundation is studied on the electro-elastic results of the size-dependent doubly curved piezoelectric shell.

Formulation of doubly curved piezoelectric shells

In this paper size-dependent electro-elastic analysis of a small scale doubly curved piezoelectric shell is presented. Figure 1 shows the schematic of a three-layered size-dependent doubly curved piezoelectric shell. The length of a material element $d s$ is defined as follows

d s^{2} = (g_{α α}) {(d α)}^{2} + (g_{β β}) {(d β)}^{2} + (g_{z z}) {(d z)}^{2}

(1)

in which

g_{α α}, g_{β β}, g_{z z}

are the components of covariant metric tensor. In addition

α, β, z

are indicated coordinates along two planar and thickness direction. Two principle radii of curvature are depicted with

R_{1}, R_{2}

. The components of covariant metric tensor are defined for coordinate system corresponding to doubly curved geometry as follows

g_{α α} = R_{1}^{2}, g_{β β} = R_{2}^{2}, g_{z z} = 1

(2)

To account size dependency and material length scale parameters, modified couple stress theory is presented for size-dependent analysis of a doubly curved piezoelectric shell resting on Pasternak's foundation in electrical environment. Based on the modified couple stress theory, the strain energy of a size-dependent structure is assumed as a function of both strain and curvature tensors as [4]

U_{s} = \frac{1}{2} \int_{v} (σ_{i j} ε_{i j} + m_{i j} χ_{i j} - D_{i} E_{i}) d V

(3)

in which σ_ij and ε_ij are the components of the stress and strain tensors, m_ij are the components of the deviatoric part of the symmetric couple stress tensor that is defined as follows [4]

m_{i j} = \frac{E}{1 + ν} l^{2} χ_{i j}

(4)

in which E is modulus of elasticity,

ν

is Poisson ratio, l is the material length scale parameter, and χ_ij are the components of the symmetric curvature tensor which is defined a

χ = \frac{1}{2} (\nabla \vec{θ} + {(\nabla \vec{θ})}^{T})

(5)

where θ_i are the components of the rotation vector related to the displacement field (

\vec{u}

= (u₁, u₂, u₃) as

\vec{θ} = \frac{1}{2} (curl (\vec{u}))

(6)

The components of rotation vector $\vec{θ}$ can be derived based on definition of curl operator in corresponding coordinate system. In the curvilinear coordinate system, the curl operator is defined as follows

\begin{array}{l} curl \vec{u} = {(g_{β β} g_{α α})}^{- \frac{1}{2}} (\frac{\partial (\sqrt{g_{z z}} u_{z})}{\partial β} - \frac{\partial (\sqrt{g_{β β}} u_{β})}{\partial z}) \vec{e_{α}} \\ + {(g_{z z} g_{α α})}^{- \frac{1}{2}} (\frac{\partial (\sqrt{g_{α α}} u_{α})}{\partial z} - \frac{\partial (\sqrt{g_{z z}} u_{z})}{\partial z}) \vec{e_{β}} \\ + {(g_{β β} g_{α α})}^{- \frac{1}{2}} (\frac{\partial (\sqrt{g_{β β}} u_{β})}{\partial α} - \frac{\partial (\sqrt{g_{α α}} u_{α})}{\partial β}) \vec{e_{z}} \end{array}

(7)

Substitution of covariant metric tensor into equation (5) leads to updated form of curl operator in curvilinear coordinate system as follows

\begin{array}{l} curl \vec{u} = \frac{1}{R_{2}} (\frac{\partial u_{z}}{\partial β} - \frac{\partial (R_{2} u_{β})}{\partial z}) \vec{e_{α}} \\ + \frac{1}{R_{1}} (\frac{\partial (R_{1} u_{α})}{\partial z} - \frac{\partial u_{z}}{\partial z}) \vec{e_{β}} \\ + \frac{1}{R_{1} R_{2}} (\frac{\partial (R_{2} u_{β})}{\partial α} - \frac{\partial (R_{1} u_{α})}{\partial β}) \vec{e_{z}} \end{array}

(8)

To arrive the basic relations of modified couple stress formulation and strain energy of system, the displacement field based on first-order shear deformation theory is expressed as follows [21]

\begin{array}{l} u_{α} = (1 + \frac{z}{R_{1}}) u + z ϕ_{α}, \\ u_{β} = (1 + \frac{z}{R_{2}}) v + z ϕ_{β}, \\ u_{z} = w \end{array}

(9)

in which

u_{α}, u_{β}, u_{z}

are displacement components along the

α, β, z

directions, respectively. In addition,

u, v, w

are displacement of middle surface and

ϕ_{α}, ϕ_{β}

are rotation components around

β, α

axes, respectively. The linear normal and shear strain components are derived in curvilinear coordinate system as follows

\begin{array}{l} ε_{α} = ε_{α}^{0} + z k_{α}, \\ ε_{β} = ε_{β}^{0} + z k_{β}, \\ ε_{β z} = ε_{β z}^{0}, \\ ε_{α z} = ε_{α z}^{0}, \\ ε_{α β} = ε_{α β}^{0} + z k_{α β} \end{array}

(10)

in which the unknown functions are expressed in Appendix 1. Substitution of equation (9) into equation (8) completes the curl operator in terms of displacement components and principle radii as follows

\begin{array}{l} curl \vec{u} = \frac{1}{R_{2}} (\frac{\partial w}{\partial β} - υ - R_{2} ϕ_{β}) \vec{e_{α}} \\ + \frac{1}{R_{1}} (u + R_{1} ϕ_{α} - \frac{\partial w}{\partial α}) \vec{e_{β}} \\ + \frac{1}{R_{1} R_{2}} (R_{2} {(1 + \frac{z}{R_{2}}) \frac{\partial v}{\partial α} + z \frac{\partial ϕ_{β}}{\partial α}} - R_{1} {(1 + \frac{z}{R_{1}}) \frac{\partial u}{\partial β} + z \frac{\partial ϕ_{α}}{\partial β}}) \vec{e_{z}} \end{array}

(11)

The gradient of rotation vector is defined in curvilinear coordinate system in terms of covariant metric tensor and components of rotation vector as follows

\begin{array}{l} \nabla \vec{θ} = [\begin{array}{c} \frac{1}{\sqrt{g_{α α} g_{β β}}} \frac{\partial (\sqrt{g_{α α}} θ_{α})}{\partial α} & \frac{1}{\sqrt{g_{α α} g_{β β}}} \frac{\partial (\sqrt{g_{α α}} θ_{α})}{\partial β} & \frac{1}{\sqrt{g_{α α}}} \frac{\partial (\sqrt{g_{α α}} θ_{α})}{\partial z} \\ \frac{1}{\sqrt{g_{α α} g_{β β}}} \frac{\partial (\sqrt{g_{β β}} θ_{β})}{\partial α} & \frac{1}{\sqrt{g_{α α} g_{β β}}} \frac{\partial (\sqrt{g_{β β}} θ_{β})}{\partial α} & \frac{1}{\sqrt{g_{β β}}} \frac{\partial (\sqrt{g_{β β}} θ_{β})}{\partial z} \\ \frac{1}{\sqrt{g_{α α}}} \frac{\partial θ_{z}}{\partial z} & \frac{1}{\sqrt{g_{β β}}} \frac{\partial θ_{z}}{\partial β} & \frac{\partial θ_{z}}{\partial z} \end{array}] \end{array}

(12)

Substitution of rotation components into the components of curvature tensor $χ = \frac{1}{2} (\nabla \vec{θ} + {(\nabla \vec{θ})}^{T})$ leads to

\begin{array}{l} χ_{11} = \frac{1}{2 R_{2}^{2}} (\frac{\partial^{2} w}{\partial β \partial α} - \frac{\partial v}{\partial α} - R_{2} \frac{\partial ϕ_{β}}{\partial α}) \\ χ_{22} = \frac{1}{2 R_{1}^{2}} (\frac{\partial u}{\partial β} + R_{2} \frac{\partial ϕ_{α}}{\partial β} - \frac{\partial^{2} w}{\partial β \partial α}) \\ χ_{33} = \frac{1}{2 R_{1} R_{2}} (\frac{\partial v}{\partial α} - \frac{\partial u}{\partial β} + R_{2} \frac{\partial ϕ_{β}}{\partial α} - R_{1} \frac{\partial ϕ_{α}}{\partial β}) \\ χ_{12} = (\frac{1}{4 R_{2}^{2}} (\frac{\partial^{2} w}{\partial β^{2}} - \frac{\partial v}{\partial β} - R_{2} \frac{\partial ϕ_{β}}{\partial β}) + \frac{1}{4 R_{1}^{2}} (\frac{\partial u}{\partial α} + R_{2} \frac{\partial ϕ_{α}}{\partial α} - \frac{\partial^{2} w}{\partial α^{2}})) \\ χ_{13} = (\frac{1}{4 R_{1}^{2} R_{2}} (R_{2} {(1 + \frac{z}{R_{2}}) \frac{\partial^{2} v}{\partial α^{2}} + z \frac{\partial^{2} ϕ_{β}}{\partial α^{2}}} - R_{1} {(1 + \frac{z}{R_{1}}) \frac{\partial^{2} u}{\partial β \partial α} + z \frac{\partial ϕ_{α}}{\partial β \partial α}})) \\ χ_{23} = (\frac{1}{4 R_{1} R_{2}^{2}} (R_{2} {(1 + \frac{z}{R_{2}}) \frac{\partial v}{\partial α \partial β} + z \frac{\partial ϕ_{β}}{\partial α \partial β}} - R_{1} {(1 + \frac{z}{R_{1}}) \frac{\partial^{2} u}{\partial β^{2}} + z \frac{\partial^{2} ϕ_{α}}{\partial β^{2}}})) \end{array}

(13)

The deviatoric part of the symmetric couple stress tensor can be derived from $m_{i j} = \frac{E}{1 + ν} l^{2} χ_{i j}$ .

\begin{array}{l} m_{11} = \frac{E}{1 + ν} l^{2} \frac{1}{2 R_{2}^{2}} (\frac{\partial^{2} w}{\partial β \partial α} - \frac{\partial v}{\partial α} - R_{2} \frac{\partial ϕ_{β}}{\partial α}) \\ m_{22} = \frac{E}{1 + ν} l^{2} \frac{1}{2 R_{1}^{2}} (\frac{\partial u}{\partial β} + R_{2} \frac{\partial ϕ_{α}}{\partial β} - \frac{\partial^{2} w}{\partial β \partial α}) \\ m_{33} = \frac{E}{1 + ν} l^{2} \frac{1}{2 R_{1} R_{2}} (\frac{\partial v}{\partial α} - \frac{\partial u}{\partial β} + R_{2} \frac{\partial ϕ_{β}}{\partial α} - R_{1} \frac{\partial ϕ_{α}}{\partial β}) \\ m_{12} = \frac{E}{1 + ν} l^{2} (\frac{1}{4 R_{2}^{2}} (\frac{\partial^{2} w}{\partial β^{2}} - \frac{\partial v}{\partial β} - R_{2} \frac{\partial ϕ_{β}}{\partial β}) + \frac{1}{4 R_{1}^{2}} (\frac{\partial u}{\partial α} + R_{2} \frac{\partial ϕ_{α}}{\partial α} - \frac{\partial^{2} w}{\partial α^{2}})) \\ m_{13} = \frac{E}{1 + ν} l^{2} (\frac{1}{4 R_{1}^{2} R_{2}} (R_{2} {(1 + \frac{z}{R_{2}}) \frac{\partial^{2} v}{\partial α^{2}} + z \frac{\partial^{2} ϕ_{β}}{\partial α^{2}}} - R_{1} {(1 + \frac{z}{R_{1}}) \frac{\partial^{2} u}{\partial β \partial α} + z \frac{\partial ϕ_{α}}{\partial β \partial α}})) \\ m_{23} = \frac{E}{1 + ν} l^{2} (\frac{1}{4 R_{1} R_{2}^{2}} (R_{2} {(1 + \frac{z}{R_{2}}) \frac{\partial v}{\partial α \partial β} + z \frac{\partial ϕ_{β}}{\partial α \partial β}} - R_{1} {(1 + \frac{z}{R_{1}}) \frac{\partial^{2} u}{\partial β^{2}} + z \frac{\partial^{2} ϕ_{α}}{\partial β^{2}}})) \end{array}

(14)

After completion of couple stress formulation, the electro-elastic relations can be completed. The constitutive relations for isotropic core and piezoelectric layers are expressed as [28 –30]

{\begin{matrix} σ_{α} \\ σ_{β} \\ σ_{β z} \\ σ_{α z} \\ σ_{α β} \end{matrix}} = {[\begin{matrix} Q_{11} & Q_{12} & 0 & 0 & 0 \\ Q_{12} & Q_{22} & 0 & 0 & 0 \\ 0 & 0 & Q_{44} & 0 & 0 \\ 0 & 0 & 0 & Q_{55} & 0 \\ 0 & 0 & 0 & 0 & Q_{66} \end{matrix}]}^{Core} {\begin{matrix} ε_{α} \\ ε_{β} \\ γ_{β z} \\ γ_{α z} \\ γ_{α β} \end{matrix}}

(15)

for core (

- \frac{h_{e}}{2} < z < \frac{h_{e}}{2}

) and

{\begin{array}{l} σ_{α} \\ σ_{β} \\ σ_{β z} \\ σ_{α z} \\ σ_{α β} \end{array}} = {[\begin{array}{c} Q_{11} & Q_{12} & 0 & 0 & 0 \\ Q_{12} & Q_{22} & 0 & 0 & 0 \\ 0 & 0 & Q_{44} & 0 & 0 \\ 0 & 0 & 0 & Q_{55} & 0 \\ 0 & 0 & 0 & 0 & Q_{66} \end{array}]}^{Layers} {\begin{array}{l} ε_{α} \\ ε_{β} \\ γ_{β z} \\ γ_{α z} \\ γ_{α β} \end{array}} - [\begin{array}{c} 0 & 0 & e_{13} \\ 0 & 0 & e_{23} \\ 0 & e_{42} & 0 \\ e_{51} & 0 & 0 \\ 0 & 0 & 0 \end{array}] {\begin{array}{l} E_{α} \\ E_{β} \\ E_{z} \end{array}}

(16)

for piezoelectric layers

(\frac{h_{e}}{2} < | z | < \frac{h_{e}}{2} + h_{p})

.In which

Q_{i j}

e_{i j}

E_{i}

are stiffness coefficients, piezoelectric coefficients, and electric fields, respectively. The electric displacement relations for piezoelectric layers are expressed as

{\begin{array}{l} D_{α} \\ D_{β} \\ D_{z} \end{array}} = [\begin{array}{c} \begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & e_{24} \\ e_{31} & e_{32} & 0 \end{array} & \begin{array}{l} e_{15} & 0 \\ 0 & 0 \\ 0 & 0 \end{array} \end{array}] {\begin{array}{l} ε_{α} \\ ε_{β} \\ γ_{β z} \\ γ_{α z} \\ γ_{α β} \end{array}} + [\begin{array}{c} \begin{array}{c} k_{11} \\ 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ k_{22} \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ k_{33} \end{array} \end{array}] {\begin{array}{l} E_{α} \\ E_{β} \\ E_{z} \end{array}}

(17)

in which

k_{i j}

are dielectric coefficients. Electric field components can be derived using electric potential distribution in terms of coordinate of doubly curved as follows [28 –30]

\begin{array}{l} \overset{⌣}{Ψ} (α, β, z) = \frac{2 z}{h} Ψ_{0} - Ψ (β,β) cos \frac{π z}{h} \end{array},

(18)

in which

Ψ_{0}

is the applied voltage. Electric field components are derived using negative gradient of electric potential distribution as follows [28 –30]

\begin{array}{l} E_{α} = - \frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial \overset{⌣}{Ψ}}{\partial α} = \frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial Ψ}{\partial α} cos \frac{π z}{h}, \\ E_{β} = - \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial \overset{⌣}{Ψ}}{\partial β} = \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial Ψ}{\partial β} cos \frac{π z}{h}, \\ E_{z} = - \frac{2}{h} Ψ_{0} - \frac{π}{h} Ψsin \frac{π z}{h} \end{array}

(19)

Substitution of electric field and strain components into stress–strain and electric displacement relations leads to

\begin{array}{l} σ_{α} = Q_{11} {\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial u}{\partial α} + \frac{w}{R_{1}} + z \frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial ϕ_{1}}{\partial α}} \\ + Q_{12} {\frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial v}{\partial β} + \frac{w}{R_{2}} + z \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial ϕ_{2}}{\partial β}} \\ σ_{β} = Q_{12} {\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial u}{\partial α} + \frac{w}{R_{1}} + z \frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial ϕ_{1}}{\partial α}} \\ + Q_{22} {\frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial v}{\partial β} + \frac{w}{R_{2}} + z \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial ϕ_{2}}{\partial β}} \\ σ_{β z} = Q_{44} {ϕ_{2} + \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial w}{\partial β} - \frac{v}{R_{2}}} \\ σ_{α z} = Q_{55} {ϕ_{1} + \frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial w}{\partial α} - \frac{u}{R_{1}}} \\ σ_{α β} = Q_{66} {\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial v}{\partial α} + \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial u}{\partial β} \\ + z [\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial ϕ_{2}}{\partial α} + \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial ϕ_{1}}{\partial β}] \\ + \frac{1}{2} (\frac{1}{R_{1}} - \frac{1}{R_{2}}) [\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial v}{\partial α} - \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial u}{\partial β}]} \end{array}

(20)

For core

((- \frac{h_{e}}{2} < z < \frac{h_{e}}{2}))

and

\begin{array}{l} σ_{α} = Q_{11} {\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial u}{\partial α} + \frac{w}{R_{1}} + z \frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial ϕ_{1}}{\partial α}} \\ + Q_{12} {\frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial v}{\partial β} + \frac{w}{R_{2}} + z \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial ϕ_{2}}{\partial β}} \\ - e_{13} {- \frac{2}{h} Ψ_{0} - \frac{π}{h} Ψsin \frac{π z}{h}} \end{array}

\begin{array}{l} σ_{β} = Q_{12} {\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial u}{\partial α} + \frac{w}{R_{1}} + z \frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial ϕ_{1}}{\partial α}} \\ + Q_{22} {\frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial v}{\partial β} + \frac{w}{R_{2}} + z \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial ϕ_{2}}{\partial β}} \\ - e_{13} {- \frac{2}{h} Ψ_{0} - \frac{π}{h} Ψsin \frac{π z}{h}} \end{array}

\begin{array}{l} σ_{β z} = Q_{44} {ϕ_{2} + \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial w}{\partial β} - \frac{v}{R_{2}}} - e_{42} {\frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial Ψ}{\partial β} cos \frac{π z}{h}} \\ σ_{α z} = Q_{55} {ϕ_{1} + \frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial w}{\partial α} - \frac{u}{R_{1}}} - e_{51} {\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial Ψ}{\partial α} cos \frac{π z}{h}} \\ σ_{α β} = Q_{66} {\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial v}{\partial α} + \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial u}{\partial β} \\ + z [\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial ϕ_{2}}{\partial α} + \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial ϕ_{1}}{\partial β}] \\ + \frac{1}{2} (\frac{1}{R_{1}} - \frac{1}{R_{2}}) [\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial v}{\partial α} - \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial u}{\partial β}]} \end{array}

\begin{array}{l} D_{α} = e_{15} {ϕ_{1} + \frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial w}{\partial α} - \frac{u}{R_{1}}} - k_{11} {\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial Ψ}{\partial α} cos \frac{π z}{h}} \\ D_{β} = e_{24} {ϕ_{2} + \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial w}{\partial β} - \frac{v}{R_{2}}} - k_{22} {\frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial Ψ}{\partial β} cos \frac{π z}{h}} \\ D_{z} = e_{31} {\frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial u}{\partial α} + \frac{w}{R_{1}} + z \frac{1}{R_{1} (1 + \frac{z}{R_{1}})} \frac{\partial ϕ_{1}}{\partial α}} \\ + e_{32} {\frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial v}{\partial β} + \frac{w}{R_{2}} + z \frac{1}{R_{2} (1 + \frac{z}{R_{2}})} \frac{\partial ϕ_{2}}{\partial β}} \\ - k_{33} {- \frac{2}{h} Ψ_{0} - \frac{π}{h} Ψsin \frac{π z}{h}} \end{array}

(21)

for piezoelectric layers

(\frac{h_{e}}{2} < | z | < \frac{h_{e}}{2} + h_{p})

Strain energy of three-layered small scale doubly curved piezoelectric shell based on couple stress formulation for isotropic core and integrated piezoelectric layers considering piezoelectric effect are defined based on the relation $U = \frac{1}{2} \int^{} (σ_{i} ε_{i} + m_{i} χ_{i} - D_{i} E_{i}) d V$ [4,6]. Substitution of components of strain, curvature, and electric field into above equations and definition of resultant component, the variation of strain energy is defined as

\begin{array}{l} δ U = \int_{α} \int_{β} [{N_{α} \frac{1}{R_{1}} \frac{\partial δ u}{\partial α} + N_{α}^{1} \frac{δ w}{R_{1}} + M_{α} \frac{1}{R_{1}} \frac{\partial δ ϕ_{1}}{\partial α}} \\ + {N_{β} \frac{1}{R_{2}} \frac{\partial δ v}{\partial β} + N_{β}^{1} \frac{δ w}{R_{2}} + M_{β} \frac{1}{R_{2}} \frac{\partial δ ϕ_{2}}{\partial β}} \\ + {N_{β z}^{1} δ ϕ_{2} + N_{β z} \frac{1}{R_{2}} \frac{\partial δ w}{\partial β} - N_{β z}^{1} \frac{δ v}{R_{2}}} + {N_{α z}^{1} δ ϕ_{1} + N_{α z} \frac{1}{R_{1}} \frac{\partial δ w}{\partial α} - N_{α z}^{1} \frac{δ u}{R_{1}}} \\ + N_{α β} \frac{1}{R_{1}} \frac{\partial δ v}{\partial α} + N_{β α} \frac{1}{R_{2}} \frac{\partial δ u}{\partial β} + M_{α β} \frac{1}{R_{1}} \frac{\partial δ ϕ_{2}}{\partial α} + M_{β α} \frac{1}{R_{2}} \frac{\partial δ ϕ_{1}}{\partial β} \\ + \frac{1}{2} (\frac{1}{R_{1}} - \frac{1}{R_{2}}) (\frac{1}{R_{1}} M_{α β} \frac{\partial δ v}{\partial α} - \frac{1}{R_{2}} M_{β α} \frac{\partial δ u}{\partial β}) \\ + {{\bar{M}}_{α} \frac{\partial^{2} δ w}{\partial α \partial β} - {\bar{M}}_{α} \frac{\partial δ v}{\partial α} - R_{2} {\bar{M}}_{α} \frac{\partial δ ϕ_{β}}{\partial α} + {\bar{M}}_{β} \frac{\partial δ u}{\partial β} - {\bar{M}}_{β} \frac{\partial^{2} δ w}{\partial α \partial β} + R_{2} {\bar{M}}_{β} \frac{\partial δ ϕ_{α}}{\partial β}} \\ + {{\bar{M}}_{z} \frac{\partial δ v}{\partial α} - {\bar{M}}_{z} \frac{\partial δ u}{\partial β} + R_{2} {\bar{M}}_{z} \frac{\partial δ ϕ_{β}}{\partial α} - R_{1} {\bar{M}}_{z} \frac{\partial δ ϕ_{α}}{\partial β}} \\ + {{\bar{M}}_{α β}^{1} \frac{\partial^{2} δ w}{\partial β^{2}} - {\bar{M}}_{α β}^{1} \frac{\partial δ v}{\partial β} - R_{2} {\bar{M}}_{α β}^{1} \frac{\partial δ ϕ_{β}}{\partial β} + {\bar{M}}_{α β}^{2} \frac{\partial δ u}{\partial α} - {\bar{M}}_{α β}^{2} \frac{\partial^{2} δ w}{\partial α^{2}} + R_{2} {\bar{M}}_{α β}^{2} \frac{\partial δ ϕ_{α}}{\partial α}} \\ + {{\bar{M}}_{α z}^{1} \frac{\partial^{2} δ v}{\partial α^{2}} + {\bar{M}}_{α z}^{2} \frac{\partial^{2} δ ϕ_{β}}{\partial α^{2}} - {\bar{M}}_{α z}^{3} \frac{\partial^{2} δ u}{\partial α \partial β} - {\bar{M}}_{α z}^{4} \frac{\partial^{2} δ ϕ_{α}}{\partial α \partial β}} \\ + {{\bar{M}}_{β z}^{1} \frac{\partial^{2} δ v}{\partial α \partial β} + {\bar{M}}_{β z}^{2} \frac{\partial^{2} δ ϕ_{β}}{\partial α \partial β} - {\bar{M}}_{β z}^{3} \frac{\partial^{2} δ u}{\partial β^{2}} - {\bar{M}}_{β z}^{4} \frac{\partial^{2} δ ϕ_{α}}{\partial β^{2}}} \\ - {\bar{D}}_{α} \frac{1}{R_{1}} \frac{\partial δ Ψ}{\partial α} - {\bar{D}}_{β} \frac{1}{R_{2}} \frac{\partial δ Ψ}{\partial β} + {\bar{D}}_{z} δ Ψ] d α d β \end{array}

(22)

in which the resultant components are defined in Appendix 2. The work done by external forces including uniform transverse loads and reaction of Pasternak's foundation is calculated as

\begin{array}{l} δ W = \int^{​} {- q [1 + \frac{h}{2 R_{1}}] [1 + \frac{h}{2 R_{2}}] + R_{f} [1 - \frac{h}{2 R_{1}}] [1 - \frac{h}{2 R_{2}}]} \end{array} R_{1} R_{2} δ w d β d α

(23)

in which the reaction of foundation is defined as:

R_{f} = K_{1} w - K_{2} \nabla^{2} w

. Arranging the variables after substitution of variation of strain energy and external works into principle of minimum potential energy

δ W - δ U

leads to the following governing equations

\begin{array}{l} δ u : \frac{\partial}{\partial α} (\frac{N_{α}}{R_{1}}) + \frac{\partial}{\partial β} (\frac{N_{β α}}{R_{2}}) + \frac{N_{α z}^{1}}{R_{1}} - \frac{1}{2} (\frac{1}{R_{1}} - \frac{1}{R_{2}}) \frac{\partial}{\partial β} (\frac{M_{β α}}{R_{2}}) \\ - \frac{\partial {\bar{M}}_{z}}{\partial β} + \frac{\partial^{2} {\bar{M}}_{α z}^{3}}{\partial α \partial β} + \frac{\partial^{2} {\bar{M}}_{β z}^{3}}{\partial β^{2}} + \frac{\partial {\bar{M}}_{β}}{\partial β} + \frac{\partial {\bar{M}}_{α β}^{2}}{\partial α} = 0 \\ δ ϕ_{1} : \frac{\partial}{\partial α} (\frac{M_{α}}{R_{1}}) - N_{α z}^{1} + \frac{\partial}{\partial β} (\frac{M_{β α}}{R_{2}}) - \frac{\partial (R_{1} {\bar{M}}_{z})}{\partial β} \\ + \frac{\partial^{2} {\bar{M}}_{α z}^{4}}{\partial α \partial β} + \frac{\partial^{2} {\bar{M}}_{β z}^{4}}{\partial β^{2}} + \frac{\partial (R_{2} {\bar{M}}_{β})}{\partial β} + \frac{\partial (R_{2} {\bar{M}}_{α β}^{2})}{\partial α} = 0 \\ δ v : + \frac{\partial}{\partial β} (\frac{N_{β}}{R_{2}}) + \frac{\partial}{\partial α} (\frac{N_{α β}}{R_{1}}) + \frac{N_{β z}^{1}}{R_{2}} + \frac{1}{2} (\frac{1}{R_{1}} - \frac{1}{R_{2}}) \frac{\partial}{\partial α} (\frac{M_{α β}}{R_{1}}) \\ \begin{array}{l} - \frac{\partial {\bar{M}}_{α}}{\partial α} + \frac{\partial {\bar{M}}_{z}}{\partial α} - \frac{\partial {\bar{M}}_{α β}^{1}}{\partial β} - \frac{\partial^{2} {\bar{M}}_{α z}^{1}}{\partial α^{2}} - \frac{\partial^{2} {\bar{M}}_{β z}^{1}}{\partial α \partial β} = 0 \end{array} \\ δ ϕ_{2} : \frac{\partial}{\partial β} (\frac{M_{β}}{R_{2}}) - N_{β z}^{1} + \frac{\partial}{\partial α} (\frac{M_{α β}}{R_{1}}) \\ - \frac{\partial (R_{2} {\bar{M}}_{α})}{\partial α} + \frac{\partial (R_{2} {\bar{M}}_{z})}{\partial α} - \frac{\partial (R_{2} {\bar{M}}_{α β}^{1})}{\partial β} - \frac{\partial^{2} {\bar{M}}_{α z}^{2}}{\partial α^{2}} - \frac{\partial^{2} {\bar{M}}_{β z}^{2}}{\partial α \partial β} = 0 \\ δ w : - \frac{N_{α}^{1}}{R_{1}} - \frac{N_{β}^{1}}{R_{2}} + \frac{\partial}{\partial β} (\frac{N_{β z}}{R_{2}}) + \frac{\partial}{\partial α} (\frac{N_{α z}}{R_{1}}) \\ - \frac{\partial^{2} {\bar{M}}_{α}}{\partial α \partial β} + \frac{\partial^{2} {\bar{M}}_{β}}{\partial α \partial β} - \frac{\partial^{2} {\bar{M}}_{α β}^{1}}{\partial β^{2}} + \frac{\partial^{2} {\bar{M}}_{α β}^{2}}{\partial α^{2}} = q [1 + \frac{h}{2 R_{1}}] [1 + \frac{h}{2 R_{2}}] \\ - R_{f} [1 - \frac{h}{2 R_{1}}] [1 - \frac{h}{2 R_{2}}] \\ δ Ψ : + \frac{\partial}{\partial α} (\frac{{\bar{D}}_{α}}{R_{1}}) + \frac{\partial}{\partial β} (\frac{{\bar{D}}_{β}}{R_{2}}) + {\bar{D}}_{z} = 0 \end{array}

(24)

The mechanical and electric resultant components can be obtained by substitution of strain and electric potential components and integration on the thickness as follows:

\begin{array}{l} {N_{α}, N_{α}^{1}, M_{α}, N_{β}, N_{β}^{1}, M_{β}} = {A_{1}, A_{8}, A_{15}, A_{4}, A_{11}, A_{18}} \frac{\partial u}{\partial α} \\ + {A_{2}, A_{9}, A_{16}, A_{5}, A_{12}, A_{19}} w \\ + {A_{3}, A_{10}, A_{17}, A_{6}, A_{13}, A_{20}} \frac{\partial ϕ_{1}}{\partial α} \\ + {A_{4}, A_{11}, A_{18}, A_{22}, A_{25}, A_{28}} \frac{\partial v}{\partial β} \\ + {A_{5}, A_{12}, A_{19}, A_{23}, A_{26}, A_{29}} w \\ + {A_{6}, A_{13}, A_{20}, A_{24}, A_{27}, A_{30}} \frac{\partial ϕ_{2}}{\partial β} \\ + {A_{7}, A_{14}, A_{21}, A_{7}, A_{14}, A_{21}} Ψ \\ + {N_{α}^{Ψ}, N_{α} {1 Ψ}^{,} M_{α}^{Ψ}, N_{β}^{Ψ}, N_{β} {1 Ψ}^{,} M_{α}^{Ψ}} \end{array}

\begin{array}{l} {N_{β z}, N_{β z}^{1}, N_{α z}, N_{α z}^{1}} = {A_{31}, A_{39}, A_{35}, A_{43}} ϕ_{2} + {A_{32}, A_{40}, A_{36}, A_{44}} \frac{\partial w}{\partial β} \\ - {A_{33}, A_{41}, A_{37}, A_{45}} v - {A_{34}, A_{42}, A_{38}, A_{46}} \frac{\partial Ψ}{\partial β}, \\ {N_{α β}, N_{β α}, M_{α β}, M_{β α}} = {A_{47}, A_{53}, A_{59}, A_{65}} \frac{\partial v}{\partial α} + {A_{48}, A_{54}, A_{60}, A_{66}} \frac{\partial u}{\partial β} \\ + {A_{49}, A_{55}, A_{61}, A_{67}} \frac{\partial ϕ_{2}}{\partial α} + {A_{50}, A_{56}, A_{62}, A_{68}} \frac{\partial ϕ_{1}}{\partial β} \\ + {A_{51}, A_{57}, A_{63}, A_{69}} \frac{\partial v}{\partial α} - {A_{52}, A_{58}, A_{64}, A_{70}} \frac{\partial u}{\partial β} \end{array}

\begin{array}{l} {{\bar{D}}_{α}, {\bar{D}}_{β}} = {A_{71}, A_{75}} ϕ_{1} + {A_{72}, A_{76}} \frac{\partial w}{\partial α} \\ - {A_{73}, A_{77}} u - {A_{74}, A_{78}} \frac{\partial Ψ}{\partial α} \\ D_{z} = A_{79} \frac{\partial u}{\partial α} + A_{80} w + A_{81} \frac{\partial ϕ_{1}}{\partial α} + A_{82} \frac{\partial v}{\partial β} \\ + A_{83} w + A_{84} \frac{\partial ϕ_{2}}{\partial β} + D_{z}^{Ψ} + A_{85} Ψ \end{array}

\begin{array}{l} {\bar{M}}_{α} = A_{86} \frac{\partial^{2} w}{\partial α \partial β} - A_{86} \frac{\partial v}{\partial α} - A_{86} R_{2} \frac{\partial ϕ_{β}}{\partial α}, \\ {\bar{M}}_{β} = A_{87} \frac{\partial u}{\partial β} - A_{87} \frac{\partial^{2} w}{\partial α \partial β} + A_{87} R_{1} \frac{\partial ϕ_{α}}{\partial β} \\ {\bar{M}}_{z} = A_{88} \frac{\partial v}{\partial α} - A_{88} \frac{\partial u}{\partial β} + A_{88} R_{2} \frac{\partial ϕ_{β}}{\partial α} - A_{88} R_{1} \frac{\partial ϕ_{α}}{\partial β} \end{array}

\begin{array}{l} {{\bar{M}}_{α β}^{1}, {\bar{M}}_{α β}^{2}} = {A_{86}, A_{88}} \frac{\partial^{2} w}{\partial β^{2}} - {A_{86}, A_{88}} \frac{\partial v}{\partial β} - R_{2} {A_{86}, A_{88}} \frac{\partial ϕ_{β}}{\partial β} \\ + {A_{88}, A_{87}} \frac{\partial u}{\partial α} + R_{2} {A_{88}, A_{87}} \frac{\partial ϕ_{α}}{\partial α} - {A_{88}, A_{87}} \frac{\partial^{2} w}{\partial α^{2}} \\ {{\bar{M}}_{α z}^{1}, {\bar{M}}_{α z}^{2}, {\bar{M}}_{α z}^{3}, {\bar{M}}_{α z}^{4}} = {A_{89}, A_{97}, A_{105}, A_{109}} \frac{\partial^{2} v}{\partial α^{2}} + {A_{90}, A_{98}, A_{106}, A_{110}} \frac{\partial^{2} ϕ_{β}}{\partial α^{2}} \\ - {A_{91}, A_{99}, A_{107}, A_{111}} \frac{\partial^{2} u}{\partial α \partial β} - {A_{92}, A_{100}, A_{108}, A_{112}} \frac{\partial^{2} ϕ_{α}}{\partial α \partial β} \\ {{\bar{M}}_{β z}^{1}, {\bar{M}}_{β z}^{2}, {\bar{M}}_{β z}^{3}, {\bar{M}}_{β z}^{4}} = {A_{93}, A_{101}, A_{105}, A_{113}} \frac{\partial^{2} v}{\partial α \partial β} + {A_{94}, A_{102}, A_{106}, A_{114}} \frac{\partial^{2} ϕ_{β}}{\partial α \partial β} \\ - {A_{95}, A_{103}, A_{107}, A_{115}} \frac{\partial^{2} u}{\partial β^{2}} - {A_{96}, A_{104}, A_{108}, A_{116}} \frac{\partial^{2} ϕ_{α}}{\partial β^{2}} \end{array}

(25)

in which the stiffness parameters are defined in Appendix 3.

Substitution of resultant components into governing equations leads to final governing equations in terms of primary displacement components as follows

\begin{array}{l} δ u : - A_{107} \frac{\partial^{4} u}{\partial β^{4}} - A_{107} \frac{\partial^{4} u}{\partial α^{2} \partial β^{2}} + (\frac{A_{1}}{R_{1}} + A_{87}) \frac{\partial^{2} u}{\partial α^{2}} \\ + (\frac{χ [A_{70} - A_{66}] + A_{54} - A_{58}}{R_{2}} + A_{88} + A_{87}) \frac{\partial^{2} u}{\partial β^{2}} \\ - \frac{A_{45}}{R_{1}} u - A_{108} \frac{\partial^{4} ϕ_{α}}{\partial α^{2} \partial β^{2}} - A_{108} \frac{\partial^{4} ϕ_{α}}{\partial β^{4}} + (\frac{A_{3}}{R_{1}} + R_{2} A_{87}) \frac{\partial^{2} ϕ_{α}}{\partial α^{2}} \\ + (\frac{A_{56} - χ A_{68}}{R_{2}} + A_{88} R_{1} + A_{87} R_{1}) \frac{\partial^{2} ϕ_{α}}{\partial β^{2}} \\ + \frac{A_{43}}{R_{1}} ϕ_{α} + A_{105} \frac{\partial^{4} v}{\partial α^{3} \partial β} + A_{105} \frac{\partial^{4} v}{\partial α \partial β^{3}} \\ + (\frac{A_{4}}{R_{1}} - A_{88} \frac{A_{57} + A_{53} - χ [A_{65} + A_{69}]}{R_{2}} - A_{88}) \frac{\partial^{2} v}{\partial α \partial β} \\ + (\frac{A_{6}}{R_{1}} + \frac{A_{55} - χ A_{67}}{R_{2}} - A_{88} R_{2} - R_{2} A_{88}) \frac{\partial^{2} ϕ_{β}}{\partial α \partial β} \\ + A_{106} \frac{\partial^{4} ϕ_{β}}{\partial α^{3} \partial β} + A_{106} \frac{\partial^{4} ϕ_{β}}{\partial α \partial β^{3}} - A_{87} \frac{\partial^{3} w}{\partial α \partial β^{2}} \\ + (A_{88} - A_{87}) \frac{\partial^{3} w}{\partial α \partial β^{2}} + (\frac{A_{5} + A_{2} + A_{44}}{R_{1}}) \frac{\partial w}{\partial α} \\ + (\frac{A_{7} - A_{46}}{R_{1}}) \frac{\partial Ψ}{\partial α} = - \frac{\partial}{\partial α} (\frac{N_{α}^{Ψ}}{R_{1}}) \end{array}

(26a)

\begin{array}{l} δ ϕ_{1} : - A_{115} \frac{\partial^{4} u}{\partial β^{4}} + A_{95} \frac{\partial^{4} u}{\partial α \partial β^{3}} + A_{91} \frac{\partial^{4} u}{\partial α^{3} \partial β} - A_{111} \frac{\partial^{4} u}{\partial α^{2} \partial β^{2}} \\ + (\frac{A_{15}}{R_{1}} + A_{87} R_{2}) \frac{\partial^{2} u}{\partial α^{2}} + (\frac{(A_{66} - A_{70})}{R_{2}} + A_{87} R_{2}) \frac{\partial^{2} u}{\partial β^{2}} \\ - (A_{88} + A_{88}) \frac{\partial^{2} u}{\partial α \partial β} - A_{116} \frac{\partial^{4} ϕ_{α}}{\partial β^{4}} - A_{112} \frac{\partial^{4} ϕ_{α}}{\partial α^{2} \partial β^{2}} + (\frac{A_{17}}{R_{1}} + A_{87} R_{2}^{2}) \frac{\partial^{2} ϕ_{α}}{\partial α^{2}} \\ + (\frac{A_{68}}{R_{2}} + A_{87} R_{1} R_{2}) \frac{\partial^{2} ϕ_{α}}{\partial β^{2}} - A_{43} ϕ_{α} + A_{109} \frac{\partial^{4} v}{\partial α^{3} \partial β} \\ + A_{113} \frac{\partial^{4} v}{\partial α \partial β^{3}} + (\frac{A_{65} + A_{69}}{R_{2}} + \frac{A_{18}}{R_{1}} - A_{88} R_{2}) \frac{\partial^{2} v}{\partial α \partial β} + A_{45} u \\ + A_{110} \frac{\partial^{4} ϕ_{β}}{\partial α^{3} \partial β} + A_{114} \frac{\partial^{4} ϕ_{β}}{\partial α \partial β^{3}} + (\frac{A_{20}}{R_{1}} + \frac{A_{67}}{R_{2}} - A_{88} R_{2}^{2}) \frac{\partial ϕ_{β}}{\partial α \partial β} - A_{87} R_{2} \frac{\partial^{3} w}{\partial α^{3}} \\ + (A_{88} R_{2} - A_{87} R_{2}) \frac{\partial^{3} w}{\partial α \partial β^{2}} + (\frac{A_{19} + A_{16}}{R_{1}} - A_{44}) \frac{\partial w}{\partial α} \\ + (\frac{A_{21}}{R_{1}} + A_{46}) \frac{\partial Ψ}{\partial α} = - \frac{\partial}{\partial α} (\frac{M_{α}^{Ψ}}{R_{1}}) \end{array}

(26b)

\begin{array}{l} δ v : + A_{91} \frac{\partial^{4} u}{\partial α^{3} \partial β} + A_{95} \frac{\partial^{4} u}{\partial α \partial β^{3}} - 2 A_{88} \frac{\partial^{2} u}{\partial α \partial β} \\ + (\frac{A_{4}}{R_{2}} + \frac{χ [A_{60} - A_{64}] + A_{48} - A_{52}}{R_{1}}) \frac{\partial^{2} u}{\partial α \partial β} + A_{92} \frac{\partial^{4} ϕ_{α}}{\partial α^{3} \partial β} \\ + A_{96} \frac{\partial^{4} ϕ_{α}}{\partial α \partial β^{3}} + (\frac{A_{50} + χ A_{62}}{R_{1}} + \frac{A_{6}}{R_{2}} - A_{88} R_{1} - R_{2} A_{88}) \frac{\partial^{2} ϕ_{α}}{\partial α \partial β} \\ - A_{89} \frac{\partial^{4} v}{\partial α^{4}} - A_{93} \frac{\partial^{4} v}{\partial α^{2} \partial β^{2}} + (\frac{A_{47} + A_{51} + χ [A_{59} + A_{63}]}{R_{1}} + A_{86} + A_{88}) \frac{\partial^{2} v}{\partial α^{2}} \\ + (A_{86} + \frac{A_{22}}{R_{2}}) \frac{\partial^{2} v}{\partial β^{2}} - \frac{A_{41}}{R_{2}} v - A_{90} \frac{\partial^{4} ϕ_{β}}{\partial α^{4}} - A_{94} \frac{\partial^{4} ϕ_{β}}{\partial α^{2} \partial β^{2}} \\ + (\frac{A_{49} + χ A_{61}}{R_{1}} + A_{86} R_{2} + A_{88} R_{2}) \frac{\partial^{2} ϕ_{β}}{\partial α^{2}} \\ + (\frac{A_{24}}{R_{2}} + R_{2} A_{86}) \frac{\partial^{2} ϕ_{β}}{\partial β^{2}} + \frac{A_{39}}{R_{2}} ϕ_{β} - A_{86} \frac{\partial^{3} w}{\partial β \partial α^{2}} - A_{86} \frac{\partial^{3} w}{\partial β^{3}} + A_{88} \frac{\partial^{3} w}{\partial α^{2} \partial β} \\ + (\frac{A_{40} + A_{5} + A_{23}}{R_{2}}) \frac{\partial w}{\partial β} + (\frac{A_{7} - A_{42}}{R_{2}}) \frac{\partial Ψ}{\partial β} = - \frac{\partial}{\partial β} (\frac{N_{β}^{Ψ}}{R_{2}}) \end{array}

(26c)

\begin{array}{l} δ ϕ_{2} : + A_{103} \frac{\partial^{4} u}{\partial α \partial β^{3}} + A_{99} \frac{\partial^{4} u}{\partial α^{3} \partial β} + (\frac{A_{18}}{R_{2}} + \frac{A_{60} - A_{64}}{R_{1}} - A_{88} R_{2} - A_{88} R_{2}) \frac{\partial^{2} u}{\partial α \partial β} + A_{41} v \\ - R_{2}^{2} A_{88} \frac{\partial^{2} ϕ_{α}}{\partial α \partial β} + A_{100} \frac{\partial^{4} ϕ_{α}}{\partial α^{3} \partial β} + A_{104} \frac{\partial^{4} ϕ_{α}}{\partial α \partial β^{3}} \\ + (\frac{A_{62}}{R_{1}} + \frac{A_{20}}{R_{2}} - A_{88} R_{1} R_{2}) \frac{\partial^{2} ϕ_{α}}{\partial α \partial β} - A_{97} \frac{\partial^{4} v}{\partial α^{4}} \\ - A_{101} \frac{\partial^{4} v}{\partial α^{2} \partial β^{2}} + (\frac{A_{59} + A_{63}}{R_{1}} + A_{86} R_{2} + A_{88} R_{2}) \frac{\partial^{2} v}{\partial α^{2}} \\ + (\frac{A_{28}}{R_{2}} + A_{86} R_{2}) \frac{\partial^{2} v}{\partial β^{2}} - A_{98} \frac{\partial^{4} ϕ_{β}}{\partial α^{4}} \\ - (A_{86} R_{2} + A_{102}) \frac{\partial^{4} ϕ_{β}}{\partial α^{2} \partial β^{2}} + (\frac{A_{30}}{R_{2}} + \frac{A_{61}}{R_{1}} + A_{86} R_{2}^{2} + A_{88} R_{2}^{2}) \frac{\partial^{2} ϕ_{β}}{\partial α^{2}} \\ + R_{2}^{2} A_{86} \frac{\partial^{2} ϕ_{β}}{\partial β^{2}} - A_{39} ϕ_{β} - A_{86} R_{2} \frac{\partial^{3} w}{\partial β^{3}} \\ + A_{88} R_{2} \frac{\partial^{3} w}{\partial α^{2} \partial β} + (\frac{A_{29} + A_{19}}{R_{2}} - A_{40}) \frac{\partial w}{\partial β} \\ + (\frac{A_{21}}{R_{2}} + A_{42}) \frac{\partial Ψ}{\partial β} = - \frac{\partial}{\partial β} (\frac{M_{α}^{Ψ}}{R_{2}}) \end{array}

(26d)

\begin{array}{l} δ w : + A_{87} \frac{\partial^{3} u}{\partial α^{3}} + (A_{87} - A_{88}) \frac{\partial^{3} u}{\partial α \partial β^{2}} + (- \frac{A_{11}}{R_{2}} - \frac{A_{8} + A_{37}}{R_{1}}) \frac{\partial u}{\partial α} \\ + (A_{87} R_{1} - R_{2} A_{88}) \frac{\partial^{3} ϕ_{α}}{\partial α \partial β^{2}} + R_{2} A_{87} \frac{\partial^{3} ϕ_{α}}{\partial α^{3}} + (\frac{A_{35} - A_{10}}{R_{1}} - \frac{A_{13}}{R_{2}}) \frac{\partial ϕ_{α}}{\partial α} \\ + A_{86} \frac{\partial^{3} v}{\partial α^{2} \partial β} + A_{86} \frac{\partial^{3} v}{\partial β^{3}} - A_{88} \frac{\partial^{3} v}{\partial α^{2} \partial β} + (- \frac{A_{33} + A_{25}}{R_{2}} - \frac{A_{11}}{R_{1}}) \frac{\partial v}{\partial β} \\ + R_{2} A_{86} \frac{\partial^{3} ϕ_{β}}{\partial β^{3}} + A_{86} R_{2} \frac{\partial^{3} ϕ_{β}}{\partial α^{2} \partial β} - R_{2} A_{88} \frac{\partial^{3} ϕ_{β}}{\partial α^{2} \partial β} \\ + (- \frac{A_{13}}{R_{1}} - \frac{A_{31}}{R_{2}} - \frac{A_{27}}{R_{2}}) \frac{\partial ϕ_{β}}{\partial β} - A_{86} \frac{\partial^{4} w}{\partial β^{4}} \\ - A_{87} \frac{\partial^{4} w}{\partial α^{4}} + (A_{88} + A_{88} - A_{86} - A_{87}) \frac{\partial^{4} w}{\partial α^{2} \partial β^{2}} \\ + \frac{A_{36}}{R_{1}} \frac{\partial^{2} w}{\partial α^{2}} + (- \frac{A_{9} + A_{12}}{R_{1}} - \frac{A_{12} + A_{26}}{R_{2}}) w \\ + (\frac{A_{32} - A_{34}}{R_{2}}) \frac{\partial^{2} Ψ}{\partial β^{2}} - \frac{A_{38}}{R_{1}} \frac{\partial^{2} Ψ}{\partial α^{2}} + (- \frac{A_{14}}{R_{2}} - \frac{A_{14}}{R_{1}}) Ψ \\ = + \frac{N_{α}^{1 Ψ}}{R_{1}} + \frac{N_{β}^{1 Ψ}}{R_{2}} \\ + (1 - μ^{2} \nabla^{2}) {q [1 + \frac{h}{2 R_{1}}] [1 + \frac{h}{2 R_{2}}] - R_{f} [1 - \frac{h}{2 R_{1}}] [1 - \frac{h}{2 R_{2}}]} \end{array}

(26e)

\begin{array}{l} δ Ψ : (- \frac{A_{73}}{R_{1}} + A_{79}) \frac{\partial u}{\partial α} + (\frac{A_{71}}{R_{1}} + A_{81}) \frac{\partial ϕ_{1}}{\partial α} \\ + (A_{82} - \frac{A_{77}}{R_{2}}) \frac{\partial v}{\partial β} + (\frac{A_{75}}{R_{2}} + A_{84}) \frac{\partial ϕ_{2}}{\partial β} + \frac{A_{72}}{R_{1}} \frac{\partial^{2} w}{\partial α^{2}} \\ + \frac{A_{76}}{R_{2}} \frac{\partial^{2} w}{\partial β^{2}} + (A_{80} + A_{83}) w - \frac{A_{78}}{R_{2}} \frac{\partial^{2} Ψ}{\partial β^{2}} \\ - \frac{A_{74}}{R_{1}} \frac{\partial^{2} Ψ}{\partial α^{2}} + A_{85} Ψ = - D_{z}^{Ψ} \end{array}

(26f)

Solution procedure

In this section solution of the governing equations is proposed. The simply-supported boundary conditions are assumed for doubly curved piezoelectric nano shell. In addition, homogeneous electrical boundary conditions are considered for piezoelectric doubly curved nano shell. For this type of boundary conditions, the Navier solution is proposed for six variables as follows

\begin{array}{l} {\begin{array}{l} u \\ ϕ_{1} \\ v \\ ϕ_{2} \\ w \\ Ψ \end{array}} = {\begin{array}{l} U c o s λ_{m} α s i n μ_{n} β \\ Φ_{1} c o s λ_{m} α s i n μ_{n} β \\ V s i n λ_{m} α c o s μ_{n} β \\ Φ_{2} s i n λ_{m} α c o s μ_{n} β \\ W s i n λ_{m} α s i n μ_{n} β \\ Ψ s i n λ_{m} α s i n μ_{n} β \end{array}} \end{array}

(27)

in which

{X} = {U, Φ_{1}, V, Φ_{2}, W, Ψ}^{T}

are unknown amplitudes and

λ_{m} = \frac{m R_{1}}{L_{1}}, μ_{n} = \frac{n R_{2}}{L_{2}}

. Substitution of proposed solution from equation (27) into governing equations of electro-elastic bending leads to the following well-known format as follows

[K] {X} = {F}

(28)

Solution of equation (28) yield the results in terms of various parameters of the problem.

Numerical results and discussion

The numerical results are presented in terms of important parameters of the problem including small scale parameters, applied voltage and two parameters of foundation. The numerical results are presented in dimensionless forms. The dimensionless displacements are introduced as

{\bar{w}, \bar{u}, \bar{v}} = \frac{{W, U, V}}{h}

In addition to account the size dependency, the small scale parameter is normalized using parameter

α

l = α h

in which

l

is small scale parameter and h is thickness of shell.

The material properties of piezoelectric doubly curved shell are assumed as

\begin{array}{l} Q_{11} = 138.499 GPa, Q_{22} = 138.499 GPa, Q_{33} = 114.745 GPa \\ Q_{12} = 77.371 GPa, Q_{13} = 73.643 GPa, Q_{23} = 73.643 GPa \\ Q_{44} = 25.6 GPa, Q_{55} = 25.6 GPa, Q_{66} = 30.6 GPa \\ e_{13} = e_{31} = - 5.2 \frac{C}{m^{2}}, e_{23} = e_{32} = - 5.2 \frac{C}{m^{2}}, e_{33} = 15.8 \frac{C}{m^{2}}, e_{15} = 12.72 \frac{C}{m^{2}} \\ e_{24} = 12.72 \frac{C}{m^{2}}, k_{11} = 1.306 \times 10^{- 8} \frac{F}{m}, k_{22} = 1.306 \times 10^{- 8} \frac{F}{m}, k_{33} = 1.151 \times 10^{- 8} \frac{F}{m} \end{array}

Shown in Figure 2 is the variation of dimensionless transverse deflection $\bar{w}$ of doubly curved piezoelectric shell in terms of dimensionless small scale parameter $α$ for various values of applied voltages $Ψ_{0}$ . It is observed that with increase of the dimensionless small scale parameter $α$ the dimensionless deflection is decreased. One can conclude that with increase of small scale parameter associated with modified couple stress theory, the stiffness of structure is increased and then the transverse deflection is decreased. Furthermore, it is observed that the deflection is increased significantly with increase of applied voltage $Ψ_{0}$ . Variation of the maximum electric potential $Ψ$ through thickness direction is observed in Figure 3. The numerical results indicate that with increase of dimensionless small scale parameter $α$ , the maximum electric potetnial is decreased. In addition, increase of applied voltage leads to significant decrease of the maximum electric potential.

The influence of two parameters of Pasternak's foundation $K_{w} . K_{G}$ on variation of the dimensionless transverse displacement $\bar{w}$ and maximum electric potential $Ψ$ are presented in Figures 4 and 5, respectively.

Figures 4 and 5 show variation of the dimensionless transverse displacement $\bar{w}$ and maximum electric potential $Ψ$ in terms of two parameters of Pasternak's foundation $K_{w} . K_{G}$ . The numerical results indicate that with the increase of both parameters of foundation, the dimensionless transverse deflection and absolute values of maximum electric potential are decreased significantly.

Two dimensional variations of significant outputs of this analysis including dimensionless in-plane and transverse deflections, rotations and maximum electric potential are presented in terms of important parameters of the problem such as small scale parameter and applied voltage in Figures 6 to 17.

Variation of dimensionless in-plane deformation $\bar{u}$ of doubly curved size-dependent piezoelectric shell is presented in terms of dimensionless small scale parameter $α$ for various applied voltages $Ψ_{0}$ in Figure 6. It is observed that this value is very small and is increased with increase of applied voltage and decrease of small scale parameter $α$ . Figure 7 shows that how rotation component $ϕ_{α}$ changes with change of dimensionless small scale parameter $α$ and various applied voltages $Ψ_{0}$ . One can conclude that this component is increased significantly with increase of applied voltages and decrease of dimensionless small scale parameter.

Figures 8 and 9 show variation of dimensionless in-plane deformation $\bar{v}$ and rotation component $ϕ_{β}$ of doubly curved piezoelectric shell in terms of small scale parameter $α$ for various applied voltages $Ψ_{0}$ . The numerical results show that the dimensionless in-plane deformation $\bar{v}$ and rotation component $ϕ_{β}$ are increased with increase of applied voltage $Ψ_{0}$ and decrease of dimensionless small scale parameter $α$ .

Figures 10 and 11 show variation of dimensionless transverse deflection $\bar{w}$ and maximum electric potential $Ψ$ of doubly curved piezoelectric shell in terms of dimensionless small scale parameter $α$ and various applied voltages $Ψ_{0}$ . The numerical results indicate that with increase of applied voltage, the transverse deflections and maximum electric potentials are increased significantly. Unlike the effect of applied voltage, the dimensionless small scale parameter $α$ leads to paradox changes on the deflections and maximum electric potential. It is observed that with the increase of small scale parameter, the deflection is decreased while the maximum electric potential is increased.

Shown in Figures 12, 13, 14, and 15 are distribution of the dimensionless displacements and rotations components $\bar{u}, ϕ_{α}, \bar{v}, ϕ_{β}$ in terms of two parameters of Pasternak's foundation, respectively. The numerical results indicate that with increase of both parameters of foundation, the stiffness of foundation is increased and consequently the displacements and rotations are decreased significantly.

The influence of two parameters of Pasternak's foundation on distribution of the dimensionless transverse deflection and maximum electric potential is investigated in Figures 16 and 17, respectively. The numerical results show that the both components are decreased with the increase of two parameters of Pasternak's foundation.

Conclusion

Modified couple stress formulation of a size dependent doubly curved piezoelectric shell subjected to applied electric potential and transverse loads resting on Pasternak's foundation was studied in this paper based on the principle of virtual work. First-order shear deformation theory was employed to derive the governing equations. An orthogonal curvilinear coordinate system was used to derive the basic geometric relations. The numerical results were presented for solution of the governing equations of a simply-supported doubly curved shell based on Navier's method. The numerical results indicate that the small scale parameter, applied voltage, and two parameters of the foundation have significant influence on the electro-elastic behavior of structure.

The effect of small scale parameter $α$ has been studied on the electro-elastic results of size-dependent doublycurved shell. The numerical results show that increase of small scale parameter leads to more rigid structure with respect to the case that this parameter is ignored. It is observed that the increase of small scale parameter leads to decrease of transverse displacement, maximum electric potential, in-plane deformations and rotations. Two parameters of Pasternak's foundation lead to significant changes of the electro-elastic results of the doublycurved piezoelectric shell. The numerical results indicate that with the increase of two parameters of foundation, all mechanical and electrical results are decreased significantly.

Footnotes

Acknowledgements

The author would also like to thank the Iranian Nanotechnology Development Committee for their ﬁnancial support.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: the University of Kashan (Grant Number: 467893/0655).

Appendix 1

References

Kapania

Yang

TY.

Formulation of an imperfect quadrilateral doubly curved shell element for postbuckling analysis. AIAA J 1986; 24: 310–311.

Fan

Zhang

Analytical solutions for thick doubly curved laminated hells. J Eng Mech 1992; 118: 7.

Tadi Beni

Mehralian

Zeighampour

The modified couple stress functionally graded cylindrical thin shell formulation. Mech Adv Mater Struct 2016; 23: 791–801.

Arefi

Kiani

Zenkour

AM.

Size-dependent free vibration analysis of a three-layered exponentially graded nano-/micro-plate with piezomagnetic face sheets resting on Pasternak’s foundation. J Sandwich Struct Mater. 2020; 22: 55–86.

Tadi Beni

Mehralian

Razavi

Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory. Compos Struct 2015; 120: 65–78.

Arefi

Zamani

Kiani

Size-dependent free vibration analysis of three-layered exponentially graded nanoplate with piezomagnetic face-sheets resting on Pasternak’s foundation. J Intell Mater Syst Struct 2018; 29: 774–786.

Chandrashekhara

Free vibrations of anisotropic laminated doubly curved shells. Comput Struct 1989; 33: 435–440.

Arefi

Zenkour

AM.

Effect of thermo-magneto-electro-mechanical fields on the bending behaviors of a three-layered nanoplate based on sinusoidal shear-deformation plate theory. J Sandwich Struct Mater. 2019; 21: 639–669.

Arefi

A complete set of equations for piezo-magnetoelastic analysis of a functionally graded thick shell of revolution. Lat Am J Solids Struct 2014; 11: 2073–2092.

10.

Qatu

Asadi

Vibration of doubly curved shallow shells with arbitrary boundaries. Appl Acoust 2012; 73: 21–27.

11.

Ghavanloo

Fazelzadeh

SA.

Free vibration analysis of orthotropic doubly-curved shallow shells based on the gradient elasticity. Compos Part B 2013; 45: 1448–1457.

12.

Shooshtari

Razavi

Linear and nonlinear free vibration of a multilayered magneto-electro-elastic doubly-curved shell on elastic foundation. Compos Part B 2015; 78: 95–108.

13.

Yazdi

AA.

Applicability of homotopy perturbation method to study the nonlinear vibration of doubly curved cross-ply shells. Compos Struct 2013; 96: 526–531.

14.

Arefi

Surface effect and non-local elasticity in wave propagation of functionally graded piezoelectric nano-rod excited to applied voltage. Appl Math Mech 2016; 37: 289–302.

15.

Zeighampour

Tadi Beni

A shear deformable cylindrical shell model based on couple stress theory. Arch Appl Mech 2015; 85: 539–553.

16.

Tornabene

Liverani

Caligiana

FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: a 2-D GDQ solution for free vibrations. Int J Mech Sci 2011; 53: 446–470.

17.

Tornabene

Fantuzzi

Bacciocchi

Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories. Compos Part B 2014; 67: 490–509.

18.

Tornabene

Fantuzzi

Bacciocchi

, et al. Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Compos Part B 2016; 89: 187–218.

19.

Arefi

Zenkour

AM.

Free vibration. wave propagation and tension analyses of a sandwich micro/nano rod subjected to electric potential using strain gradient theory. Mater Res Express 2016; 3: 115704.

20.

Tornabene

Fantuzzi

Bacciocchi

, et al. MLSDQ based on RBFs for the free vibrations of laminated composite doubly-curved shells. Compos Part B 2016; 99: 30–47.

21.

Pouresmaeeli

Fazelzadeh

SA.

Frequency analysis of doubly curved functionally graded carbon nanotube-reinforced composite panels. Acta Mech 2016; 227: 2765–2794.

22.

Arefi

Zenkour

AM.

Employing the coupled stress components and surface elasticity for nonlocal solution of wave propagation of a functionally graded piezoelectric Love nanorod model. J Intell Mater Syst Struct 2017; 28: 2403–2413.

23.

Zhou

Wang

Vibration and stability of micro-scale cylindrical shells conveying fluid based on modified couple stress theory.

IET Micro Nano Lett 2012; 7.

24.

Arefi

Zenkour

AM.

Transient sinusoidal shear deformation formulation of a size-dependent three-layer piezo-magnetic curved nanobeam. Acta Mech 2017; 228: 3657–3674.

25.

Arefi

Zenkour

AM.

Influence of magneto-electric environments on size-dependent bending results of three-layer piezomagnetic curved nanobeam based on sinusoidal shear deformation theory. J Sandwich Struct Mater. 2019; 21: 2751–2778.

26.

Arefi

Zenkour

AM.

Thermal stress and deformation analysis of a size-dependent curved nanobeam based on sinusoidal shear deformation theory. Alexandria Eng J. 2018; 57: 2177–2185.

27.

Arefi

Zenkour

AM.

Electro-magneto-elastic analysis of a three-layer curved beam. Smart Struct Syst 2017; 19: 695–703.

28.

Zeighampour

Shojaeian

Buckling analysis of functionally graded sandwich cylindrical micro/nanoshells based on the couple stress theory. J Sandwich Struct Mater. 2019; 21: 917–937.

29.

Arefi

Zenkour

AM.

Wave propagation analysis of a functionally graded magneto-electro-elastic nanobeam rest on Visco-Pasternak foundation. Mech Res Commun 2017; 79: 51–62.

30.

Arefi

Zenkour

AM.

Size-dependent vibration and bending analyses of the piezomagnetic three-layer nanobeams. Appl Phys A 2017; 123: 202.

31.

Shah

Batra

RC.

Stretching and bending deformations due to normal and shear tractions of doubly curved shells using third-order shear and normal deformable theory . Mech Adv Mater Struct 2018; 15–16: 1276–1296.

32.

Kar

Panda

Mahapatra

TR.

Thermal buckling behaviour of shear deformable functionally graded single/doubly curved shell panel with TD and TID properties. Adv Mater Res 2016; 5: 205–221.

33.

Liew

Lim

CW.

A higher-order theory for vibration of doubly curved shallow shells. J Appl Mech 1996; 63: 587–593.

34.

Coşkun I. The response of a finite beam on a tensionless Pasternak foundation subjected to a harmonic load. Eur J Mech A/Solids 2003; 22: 151–161.

35.

Szekrényes

Improved analysis of unidirectional composite delamination specimens. Mech Mater 2007; 39: 953–974.

36.

Arefi

Analysis of wave in a functionally graded magneto-electro-elastic nano-rod using nonlocal elasticity model subjected to electric and magnetic potentials. Acta Mech 2016; 227: 2529–2542.